In this section, we introduce an alternative viz. Effective Lyapunov Exponent (ELE) for the classical Lyapunov exponent.
We will follow the notation
and definition used by
Deshpande and Gejji [1]}.
They define the fractional equivalent for the
\(x(n+1)=f(x(n))\)  in the following manner.
They construct an discrete Caputo-type fractional
difference operator and define
\(u(t)=u_0 + {\frac{1}{\Gamma (\alpha )}} \sum _{j=1}^{t}{\frac{\Gamma (t-j+\alpha )}{\Gamma (t-j+1)}}\left( f(j-1,u(j-1))-u(j-1)\right)\) 
in general. We assume that
the function \(f\)  does not depend on time. We define
\(g_{\alpha }(k)= {\frac{\Gamma (k+\alpha )}{\Gamma (k+1)}}\) 
and alternatively write above expression as
\(u(t)=u_0 + {\frac{1}{\Gamma (\alpha )}} \sum _{j=1}^{t}g_{\alpha }(t-j) \left(f(u(j-1))-u(j-1)\right)\) .
